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**Published in:** LIPIcs, Volume 217, 25th International Conference on Principles of Distributed Systems (OPODIS 2021)

We consider the fundamental problem of assigning distinct labels to agents in the probabilistic model of population protocols. Our protocols operate under the assumption that the size n of the population is embedded in the transition function. Their efficiency is expressed in terms of the number of states utilized by agents, the size of the range from which the labels are drawn, and the expected number of interactions required by our solutions. Our primary goal is to provide efficient protocols for this fundamental problem complemented with tight lower bounds in all the three aspects. W.h.p. (with high probability), our labeling protocols are silent, i.e., eventually each agent reaches its final state and remains in it forever, and they are safe, i.e., never update the label assigned to any single agent. We first present a silent w.h.p. and safe labeling protocol that draws labels from the range [1,2n]. Both the number of interactions required and the number of states used by the protocol are asymptotically optimal, i.e., O(n log n) w.h.p. and O(n), respectively. Next, we present a generalization of the protocol, where the range of assigned labels is [1,(1+ε) n]. The generalized protocol requires O(n log n / ε) interactions in order to complete the assignment of distinct labels from [1,(1+ε) n] to the n agents, w.h.p. It is also silent w.h.p. and safe, and uses (2+ε)n+O(n^c) states, for any positive c < 1. On the other hand, we consider the so-called pool labeling protocols that include our fast protocols. We show that the expected number of interactions required by any pool protocol is ≥ (n²)/(r+1), when the labels range is 1,… , n+r < 2n. Furthermore, we provide a protocol which uses only n+5√ n +O(n^c) states, for any c < 1, and draws labels from the range 1,… ,n. The expected number of interactions required by the protocol is O(n³). Once a unique leader is elected it produces a valid labeling and it is silent and safe. On the other hand, we show that (even if a unique leader is given in advance) any silent protocol that produces a valid labeling and is safe with probability > 1-(1/n), uses ≥ n+√{(n-1)/2}-1 states. Hence, our protocol is almost state-optimal. We also present a generalization of the protocol to include a trade-off between the number of states and the expected number of interactions. Finally, we show that for any silent and safe labeling protocol utilizing n+t < 2n states, the expected number of interactions required to achieve a valid labeling is ≥ (n²)/(t+1).

Leszek Gąsieniec, Jesper Jansson, Christos Levcopoulos, and Andrzej Lingas. Efficient Assignment of Identities in Anonymous Populations. In 25th International Conference on Principles of Distributed Systems (OPODIS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 217, pp. 12:1-12:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gasieniec_et_al:LIPIcs.OPODIS.2021.12, author = {G\k{a}sieniec, Leszek and Jansson, Jesper and Levcopoulos, Christos and Lingas, Andrzej}, title = {{Efficient Assignment of Identities in Anonymous Populations}}, booktitle = {25th International Conference on Principles of Distributed Systems (OPODIS 2021)}, pages = {12:1--12:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-219-8}, ISSN = {1868-8969}, year = {2022}, volume = {217}, editor = {Bramas, Quentin and Gramoli, Vincent and Milani, Alessia}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2021.12}, URN = {urn:nbn:de:0030-drops-157871}, doi = {10.4230/LIPIcs.OPODIS.2021.12}, annote = {Keywords: population protocol, state efficiency, time efficiency, one-way epidemics, leader election, agent identities} }

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**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

We consider Boolean circuits over {or, and, neg} with negations applied only to input variables. To measure the "amount of negation" in such circuits, we introduce the concept of their "negation width". In particular, a circuit computing a monotone Boolean function f(x_1,...,x_n) has negation width w if no nonzero term produced (purely syntactically) by the circuit contains more than w distinct negated variables. Circuits of negation width w=0 are equivalent to monotone Boolean circuits, while those of negation width w=n have no restrictions. Our motivation is that already circuits of moderate negation width w=n^{epsilon} for an arbitrarily small constant epsilon>0 can be even exponentially stronger than monotone circuits.
We show that the size of any circuit of negation width w computing f is roughly at least the minimum size of a monotone circuit computing f divided by K=min{w^m,m^w}, where m is the maximum length of a prime implicant of f. We also show that the depth of any circuit of negation width w computing f is roughly at least the minimum depth of a monotone circuit computing f minus log K. Finally, we show that formulas of bounded negation width can be balanced to achieve a logarithmic (in their size) depth without increasing their negation width.

Stasys Jukna and Andrzej Lingas. Lower Bounds for DeMorgan Circuits of Bounded Negation Width. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 41:1-41:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{jukna_et_al:LIPIcs.STACS.2019.41, author = {Jukna, Stasys and Lingas, Andrzej}, title = {{Lower Bounds for DeMorgan Circuits of Bounded Negation Width}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {41:1--41:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.41}, URN = {urn:nbn:de:0030-drops-102801}, doi = {10.4230/LIPIcs.STACS.2019.41}, annote = {Keywords: Boolean circuits, monotone circuits, lower bounds} }

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**Published in:** LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)

We consider normalized Boolean circuits that use binary operations of disjunction and conjunction, and unary negation, with the restriction that negation can be only applied to input variables. We derive a lower bound trade-off between the size of normalized Boolean circuits computing Boolean semi-disjoint bilinear forms and their conjunction-depth (i.e., the maximum number of and-gates on a directed path to an output gate). In particular, we show that any normalized Boolean circuit of at most epsilon log n conjunction-depth computing the n-dimensional Boolean vector convolution has Omega(n^{2-4 epsilon}) and-gates. Analogously, any normalized Boolean circuit of at most epsilon log n conjunction-depth computing the n x n Boolean matrix product has Omega(n^{3-4 epsilon}) and-gates. We complete our lower-bound trade-offs with upper-bound trade-offs of similar form yielded by the known fast algebraic algorithms.

Andrzej Lingas. Small Normalized Boolean Circuits for Semi-disjoint Bilinear Forms Require Logarithmic Conjunction-depth. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 26:1-26:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{lingas:LIPIcs.CCC.2018.26, author = {Lingas, Andrzej}, title = {{Small Normalized Boolean Circuits for Semi-disjoint Bilinear Forms Require Logarithmic Conjunction-depth}}, booktitle = {33rd Computational Complexity Conference (CCC 2018)}, pages = {26:1--26:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-069-9}, ISSN = {1868-8969}, year = {2018}, volume = {102}, editor = {Servedio, Rocco A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.26}, URN = {urn:nbn:de:0030-drops-88630}, doi = {10.4230/LIPIcs.CCC.2018.26}, annote = {Keywords: Boolean circuits, semi-disjoint bilinear form, Boolean vector convolution, Boolean matrix product} }

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